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In this study, a thermomechanical rigid-flexible coupling vibration model of the rotating pre-twisted porous functionally graded (PP-FG) thick microplate in a thermal environment is established based on the third-order shear deformation theory (TSDT) and modified couple stress theory (MCST). Three thermal distributions are considered. The effects of temperature and porosity on the equivalent material parameters are accounted for in the modified Voigt rule of mixture. The governing equations of motion are derived using the Euler–Lagrange equation and numerically solved through the complex modal analysis method and the Chebyshev–Ritz method. After validating the convergence and accuracy of the proposed model, the effects of temperature, material length scale parameter (MLSP), rotational speed, porosity index, gradient index, presetting angle, and pre-twist angle on the vibration behavior of microplates were examined. The key findings of the study are as follows: (1) The effects of rotational speed on in-plane and out-of-plane frequencies are opposite. Temperature elevation weakens the size effect and centrifugal stiffening effect on the frequencies. (2) The effects of the porosity index on the frequencies are the opposite when the gradient index exceeds a specific critical value. (3) The effects of the presetting angle on the frequencies are periodic and symmetrical at about 0. The frequencies reach an extreme value when the pre-twist angle is approximately 2$\pi$/3 for the cantilever microplate. The proposed model is more sophisticated and comprehensive than others reported in the literature. It offers a more thorough analysis and insight into the behavior and performance of micro-electro-mechanical systems (MEMS) and micro air vehicles (MAVs), particularly when operating in challenging conditions.

The cantilevered micro-beam is an important component in the field of Atomic Force Microscope (AFM) and its behavior is governed by chaotic dynamics. To better understand these dynamics, a theoretical model based on the Euler–Bernoulli beam model has been developed and experiments have been conducted. The model incorporates both the modified couple stress theory (MCST) and external forces to account for the size-dependent effect and electrostatic load. The reduced model in one degree of freedom is obtained by the Galerkin method. The validity of the reduced model is tested by comparing the results from simulations and published results and experiments. The Melnikov method is then used to detect the chaotic threshold the cantilevered micro-beam. The numerical results test the effects of MCST on the complex responses of the micro-cantilever. Implications of the research can extend to the design and optimization of MEMS devices.

This study centers on the analysis of a scale-dependent microplate configuration characterized by a porous core enveloped by nanocomposite patches embedded with graphene nanoplatelets. The microplate’s behavior is explored within a humid environment to comprehend the effects of moisture variations on its dynamic performance. Additionally, the microplate rests upon a Kerr foundation, a three-parameter elastic substrate. The material properties of all three layers are contingent upon thickness. To enhance precision, an innovative quasi-three-dimensional shear and normal trigonometric theory is employed, elucidating the kinematic interrelations of the microstructure. Notably, this novel theory accommodates the presence of transverse normal strain. For a comprehensive analysis of size influences, the modified couple stress theory is harnessed. This theory integrates a material length-scale parameter to anticipate outcomes at the micro-scale. By invoking Hamilton’s principle, differential motion equations are deduced and subsequently solved analytically. The investigation centers on probing the consequences of varied parameters on the natural frequencies. The findings underscore that the incorporation of GPLs amplifies the microplate’s stiffness, thus elevating its natural frequencies. In contrast, an escalation in the porosity index leads to a reduction in natural frequencies.

This study developed a rigid-flexible coupling vibration model for a rotating functionally graded geometrically imperfect microbeam, integrating refined shear deformation theory (RSDT), the von Kármán nonlinear assumption, and modified couple stress theory. The material properties are described by a power-law distribution along the thickness. Utilizing RSDT, deformation was characterized by axial tensile, chordwise bending, and shear variables. The energy of the microbeam was discretized through assumed modal functions that satisfy cantilever beam boundary conditions, and the governing equations were derived using the Lagrange equation, subsequently solved with the complex modal space method. After validating the model, the effects of shear and geometric imperfections on dimensionless natural frequencies and mode shapes were analyzed, taking into account dynamic stiffening, gradient index, and size effects. Key findings include: (i) shear and imperfections reduced the rotational speed, gradient index, and material length scale required for frequency locus veering; (ii) dynamic stiffening and size effects suppressed the influence of shear and imperfections; (iii) larger imperfections initially increased, then decreased, their impact on chordwise deformation modes as the gradient index increases; (iv) coupling between chordwise and axial motions leads to mode transition with increasing imperfection amplitude. This work provided theoretical guidance for addressing the disorder problem of system parameters in micro-electromechanical systems-based devices.

The stepped microbeams are typical low stiffness structures widely used in MEMS devices. Approximated solutions of the natural frequency and the pull-in voltage of stepped microbeams, including clamped-free (CF) and clamped–clamped (CC) boundary conditions are developed, and the unique pull-in behavior of the stepped microbeams is investigated. The stepped microbeam is viewed as a beam with a rectangular electrode pad at its tip for CF beam and at its center for CC beam. The motion equations are deduced based on the Euler–Bernoulli beam and the modified couple stress theory. The natural frequency and the pull-in voltage are extracted with a one-degree-of-freedom model. The present model correctness is validated by comparing with the finite element results, and the effect of the length ratio of the electrode pad to the beam and the width ratio of the beam to the electrode pad are discussed. The results show that both the natural frequency and the pull-in voltage monotonously increase with the increase of the width ratio, and first decrease and then increase with the increase of the length ratio. The minimum value of them does not appear at the same time, but is determined by the width ratio. The results can be used to design and improve the performance of MEMS devices.

A nonclassical first-order shear deformation shell model is developed to analyze the axial buckling and dynamic stability of microshells made of functionally graded materials (FGMs). For this purpose, the modified couple stress elasticity theory is implemented into the first-order shear deformation shell theory. Unlike the classical shell theory, the newly developed shell model contains an internal material length scale parameter to capture efficiently the size effect. By using the Hamilton's principle, the higher-order governing equations and boundary conditions are derived. Afterward, the Navier solution is utilized to predict the critical axial buckling loads of simply-supported functionally graded (FG) microshells. Moreover, the governing equations are written in the form of Mathieu–Hill equations and then Bolotin's method is employed to determine the instability regions. A parametric study is conducted to investigate the influences of static load factor, axial wave number, dimensionless length scale parameter, material property gradient index, length-to-radius and length-to-thickness aspect ratios on the axial buckling and dynamic stability responses of FGM microshells. It is revealed that size effect plays an important role in the value of critical axial buckling load and instability region of FGM microshells especially corresponding to those with lower aspect ratios.

This paper is concerned with the flexural vibration of an atomic force microscope (AFM) cantilever. The cantilever problem is formulated on the basis of the modified couple stress theory and the Timoshenko beam theory. The modified couple stress theory is a nonclassical continuum theory that includes one additional material parameter to describe the size effect. By using the Hamilton's principle, the governing equation of motion and the boundary conditions are derived for the AFM cantilevers. The equation is solved using the differential quadrature method for the natural frequencies and mode shapes. The effects of the sample surface contact stiffness, length scale parameter and location of the sensor tip on the flexural vibration characteristics of AFM cantilevers are discussed. Results show that the size effect on the frequency is significant when the thickness of the microcantilever has a similar value to the material length scale parameter.

In this study, the free vibration analysis of edge cracked cantilever microscale beams composed of functionally graded material (FGM) is investigated based on the modified couple stress theory (MCST). The material properties of the beam are assumed to change in the height direction according to the exponential distribution. The cracked beam is modeled as a modification of the classical cracked-beam theory consisting of two sub-beams connected by a massless elastic rotational spring. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new nonclassical beam model reduces to the classical one when the length scale parameter is zero. The problem considered is investigated using the Euler–Bernoulli beam theory by the finite element method. The system of equations of motion is derived by Lagrange’s equations. To verify the accuracy of the present formulation and results, the frequencies obtained are compared with the results available in the literature, for which good agreement is observed. Numerical results are presented to investigate the effect of crack position, beam length, length scale parameter, crack depth, and material distribution on the natural frequencies of the edge cracked FG microbeam. Also, the difference between the classical beam theory (CBT) and MCST is investigated for the vibration characteristics of the beam of concern. It is believed that the results obtained herein serve as a useful reference for research of similar nature.

This paper studies the size-dependent dynamic behavior of electrostatically actuated microaccelerometers using the modified couple stress theory. The device is modeled as a cantilevered microbeam with an electrostatically actuated proof mass attached to its free end. The equation of motion is derived based on the Hamilton’s principle and solved both numerically (using the finite element and finite difference methods) and analytically (using the perturbation technique) and the dynamic response and pull-in instability of the device is studied. The results of these methods are compared and the source of error in the analytical results at high values of external acceleration is discussed. Furthermore, the results are compared with those evaluated based on the classical theory. It is found that for cantilevered accelerometers with a beam thickness of the order of the material length scale parameter, the classical theory gives a rough estimation of the dynamic response of the system. In this situation, the error of using the classical theory may change the prediction of the system behavior from unstable to stable.

Some high-speed rotating micro-machines and micro-vibration devices rely on the use of whirling micro-shafts subject to the effect of gravity and magnetic fields. At present, the consequences of the interaction between the elastic deformation of such shafts and the magnetic/gravitational field effects remain unresolved. Focusing on micro-scale whirling shafts with very high torsional rigidity, this study presents a theoretical treatment grounded in the theory of micro-continuum elasticity to examine the ramification of this interaction. The differential transformation method (DTM) is used to obtain extensive numerical results for qualitative assessments of the magnetic-gravitational effects interaction on standing, hanging and horizontally positioned spinning micro-scale shafts. The influence of bearing-support flexibility on the response of the whirling micro-shaft is also considered with rotational and translational springs. The gravitational sag reduces the stability of whirling standing micro-shafts and increases that of the hanging micro-shafts. Further, for all the micro-shafts configurations investigated, the magnetic field is observed to stiffen the response of the shaft and favorably shifts the critical points of vibration of the whirling shafts forward.

The size-dependent dynamic behavior and flexural vibration of rotating micro-rings are investigated in this paper. Using the modified couple stress theory and Hamilton’s principle, the governing equations of motion of the rotating micro-ring are derived. The natural frequencies for both extensional and inextensional micro-rings are obtained in closed form along with the forward and backward traveling waves derived. The results indicate that the natural frequencies of the rotating micro-ring are clearly size dependent, but the size dependency decreases as the speed of rotation of the ring increases, while it decreases when the radius-to-thickness ratio of the ring increases. A comparison between the natural frequencies of the extensional and inextensional micro-rings is performed. Moreover, the effect of the radius-to-thickness ratio of the ring on the behavior of the micro-ring is investigated. Good agreement is found between the natural frequencies obtained and the experimental results reported in the literature.

In this paper, the dynamic and nonlinear vibration responses of a microresonator containing a microbridge with a proof mass located at its middle are studied. The proof mass of the microresonator is actuated by the electrostatic field in such a way that a direct voltage finds a certain equilibrium position and then be prompted to vibration under the alternative voltage. Due to the importance of the size dependency effect in analysis of the performance of microelectromechanical systems, the size dependent theory is used in the modeling of the microstructure. By adopting the modified couple stress theory and considering electrostatic actuation, the dynamic equation of motion is derived using the extended Hamilton’s principle. Further, with the approximation by Galerkin’s method, the governing equation for the static and oscillatory motion is reduced and the resultant equation is solved by analytical (multiple-scales) and numerical methods. In the analytical and numerical results, the effects of various parameters on the system response, including the midplane stretching and size dependent effects, and dependency of vibration response to initial conditions, are analyzed in detail.

This paper is concerned with the buckling and post-buckling behaviors of a simply supported symmetric functionally graded (FG) microplate lying on a nonlinear elastic foundation. The modified couple stress theory is used to capture the size effects of the FG microplate, and the Mindlin plate theory with von Karman’s geometric nonlinearity taken into account is adopted to describe its deflection behavior. Based on these assumptions and the principle of minimum potential energy, the equilibrium equations of the FG microplate and associated boundary conditions are derived. By applying the Galerkin method to the equilibrium equations, closed-form solutions for the critical buckling load and the load–displacement relation in the post-buckling stage are obtained. Furthermore, the effects of the power law index, the material length scale parameter to thickness ratio, the stiffness of the elastic foundation, and in-plane boundary conditions on the buckling and post-buckling behaviors of the FG microplate are discussed in detail.

Based on the modified couple stress theory, an attempt is made in this study to analyze the nonlinear snap-through instability of shallow sandwich arches. The microstructure-dependent functionally graded material (FGM) arch with surface bonded piezoelectric actuator layers is analyzed. The piezo-FGM sandwich arch is subjected to uniform transverse pressure load in thermo-electrical environment. All material properties of the FGM micro arch are assumed to be temperature- and position-dependent. The governing equilibrium equations of the piezo-FGM sandwich arch are established with the aid of virtual displacement principle and the uncoupled thermoelacticity theory. The obtained governing differential equations are based on the first-order shear deformation shallow arch theory of the Timoshenko and von Kármán nonlinear assumptions. These equilibrium equations contain three coupled ordinary differential equations in terms of displacements. The nondimensional governing equations are solved for the cases of piezo-FGM sandwich arches with simply supported and clamped boundary conditions by using the two-step perturbation technique. Analytical closed-form solutions are derived to give the deflected shape of the piezo-FGM sandwich arch with immovable ends. Comparison is made with the existing results for the cases of FGM arch without couple stress and piezoelectric layers, where good agreement is obtained. The nonlinear behavior of the sandwich arches is highly affected by the couple stress, piezoelectric layers, temperature change, volume fraction index, and geometrical properties of the arch.

In this study, the large amplitude free vibration of nanobeams based on the modified couple stress theory was developed by using Total Lagrangian finite element formulation. In this study, Timoshenko beam theory has been used in free vibration analysis of nanobeams. Minimal kinematic assumptions have been used to model nanobeams. With this model, free vibration of nanobeams with small to large amplitude and with arbitrary boundary conditions can be analyzed. The numerical results obtained for free vibration based on the modified couple stress theory with small amplitude and the results obtained for free vibration with large amplitude without considering the modified couple stress theory are in good agreement with the similar results reported in previous research. Effects of the dimensionless length scale parameter, slenderness ratio, vibration amplitude and different boundary conditions on the nonlinear frequency ratio of nanobeams have been investigated. The results show the importance of considering nonlinear and size effects in the free vibration analysis of nanobeams with large amplitude.

This paper analyzes the dynamic stability of an isotropic viscoelastic Euler–Bernoulli nano-beam using piezoelectric materials. For this purpose, the size-dependent theory was used in the framework of the modified couple stress theory (MCST) for piezoelectric materials. In order to capture the geometrical nonlinearity, the von Karman strain displacement relation was applied. Hamilton’s principle was also employed to obtain the governing equations. Furthermore, the Galerkin method was used in order to convert the governing partial differential equations (PDEs) to a nonlinear second-order ordinary differential one. Dynamic stability analysis was performed and the effects of such parameters as viscoelastic coefficients, size effect, and piezoelectric coefficient were investigated. The results showed that in this system, saddle points, central points, Hopf bifurcation points, and fork bifurcation points could be created, and the phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, and homoclinic orbits.

Micro air vehicles, which are typical small-sized rotating-motion systems, have seen major advancements in recent years. To provide some theoretical basis for developing and producing micro air vehicles, this study establishes a novel rigid–flexible coupling dynamic model for functionally graded (FG) moderately thick rectangular microplates attached to a central rotating rigid hub based on the modified couple stress theory and first-order shear deformation theory. The proposed model incorporates nonlinear coupling term of in-plane deformation to reflect the dynamic stiffening effect caused by rotational motion. Material characteristics of the FG microplate have a linear power-law distribution along the thickness axis. Further, the discrete form dimensionless coupling dynamic equations and their numerical solutions are obtained by combining the Euler–Lagrange equation and the Chebyshev–Ritz method. Convergence and comparative studies are carried out to demonstrate the accuracy and validity of the proposed model. Thereafter, the influence of material length scale parameter, rotational speed, gradient index, and aspect ratio on the frequency of the microplates is investigated. Numerical results reveal that couple stress and dynamic stiffening effects both enhance the rigidity of the microplates, whereas the gradient index decreases the rigidity. Nonlinear coupling term which leads to significant differences in frequency value and trace line can’t be ignored for rotative structure. In-plane motion and its coupling terms play a significant function for the moderately thick or thick microplates. The increase of rotational speed and gradient index will reduce the size dependency of the microplate. Furthermore, the frequency trajectory steering and corresponding mode transition phenomenon are graphically represented.

Rotating machinery with flexible shafts finds application across a broad spectrum, ranging from everyday household appliances to heavy-duty industrial setups. These machines harness substantial rotational energy, which in turn induces vibrations. In step with industrial progress, modern devices are becoming smart, intelligent, and compact, with the help of microscopic devices known as Micro-Electro-Mechanical Systems (MEMS), incorporating electrical and moving mechanical parts. One noteworthy category within MEMS is Power MEMS. Operating at speeds exceeding a million revolutions per minute, these systems are employed in compact energy supply solutions for small-scale electronics, diminutive ground robots and unmanned airborne vehicles, all of which demand efficient power sources. This paper addresses the rotor dynamics associated with micro-rotating systems. The intricate dynamics and nonlinear issues witnessed in macro-scale systems are equally relevant to these tiny systems. This is why recognizing the traits and behavior of these small-scale systems and analyzing their dynamic actions within the operational context becomes a fundamental topic that influences design, control, maintenance, and safety considerations. Incorporating low friction bearings and rotor dynamics into mathematical formulation yields results that are nearly perfect models. Therefore, this study aims to conduct a comprehensive nonlinear analysis of the micro-rotor, considering the nonlinearity arising from the substantial deformation of the shaft. The influence of small-scale effects, which hold significance at the micron scale, is addressed utilizing modified couple stress theory. The study employs the Euler–Bernoulli beam theory while accounting for the axial stretching effect under conditions of significant deformation. The dynamics of the rotor, including essential parameters like spin speed, disk placement, and size dependency, are thoroughly investigated through a comprehensive parametric study presented in this work.

The present study examines a microplate with a porous structure and two nanocomposite piezoelectric layers. All the layers’ properties are graded functionally, bonded to each other, and supported by an elastic foundation that can withstand both normal and shear loads. Additionally, carbon nanotubes are used to increase the electro-mechanical performance of the piezoelectric patches, which are exposed to an externally applied electric voltage. Using a higher-order trigonometric shear deformation theory and von Karman’s assumptions, the kinematic relations are demonstrated. The governing motion equations are derived using Hamilton’s principle and variational technique, and the modified couple stress theory is employed to take the scale effect into account. An analytical method based on Fourier series functions is used to solve the differential motion equations, and the impact of diverse factors such as porosity percentage, pore distribution patterns, carbon nanotubes distribution patterns, and other key parameters on the normalized frequencies of the model is analyzed after verifying the accuracy of the results. The findings of this research may aid in the development and production of smart structures and devices with increased efficiency.

This paper presents a novel shear deformation theory for analyzing porous microbeams’ bending, buckling, and free vibration resting on a foundation. The proposed shear function incorporating three kinetic variables satisfies zero-traction boundary conditions on the top and bottom surfaces of the beams and does not require a shear correction factor. The modified couple stress theory accounts for the size-dependent effects, and the governing equations are derived from Lagrange’s equation using the proposed shear function. Legendre–Ritz functions are developed to analyze the porous microbeams’ buckling, free vibration, and bending behaviors. The effects of material length scale parameter, porosity, span-to-height ratio, boundary condition, and foundation parameter on the mechanical responses of beams are investigated. Numerical results demonstrate the accuracy and efficiency of the proposed theory and can serve as benchmarks for future analysis of porous microbeams on elastic foundations.